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A complexity analysis of the F4 Gröbner basis algorithm with tracer data

Robin Kouba, Vincent Neiger, Mohab Safey El Din

Abstract

We provide a new complexity bound for the computation of grevlex Gröbner bases in the generic zero-dimensional case, relying on Moreno-Socías' conjecture. We first formalize a property of regular sequences that implies a well-known folklore consequence, which we call the increasing degree property. We then derive a new understanding of the selection of pairs in the F4 algorithm based on Moreno-Socías' conjecture. Moreover, we obtain an exact formula for the number of elements in the grevlex Gröbner basis of a given degree, for half of the relevant degrees. Combining these results, we derive a precise complexity formula for the F4 Tracer algorithm, together with its asymptotic behavior when the number of variables tends to infinity. These results yield an improvement over the state-of-the-art complexity bounds by a factor which is exponential in the number of variables.

A complexity analysis of the F4 Gröbner basis algorithm with tracer data

Abstract

We provide a new complexity bound for the computation of grevlex Gröbner bases in the generic zero-dimensional case, relying on Moreno-Socías' conjecture. We first formalize a property of regular sequences that implies a well-known folklore consequence, which we call the increasing degree property. We then derive a new understanding of the selection of pairs in the F4 algorithm based on Moreno-Socías' conjecture. Moreover, we obtain an exact formula for the number of elements in the grevlex Gröbner basis of a given degree, for half of the relevant degrees. Combining these results, we derive a precise complexity formula for the F4 Tracer algorithm, together with its asymptotic behavior when the number of variables tends to infinity. These results yield an improvement over the state-of-the-art complexity bounds by a factor which is exponential in the number of variables.
Paper Structure (20 sections, 35 theorems, 222 equations, 7 figures, 4 algorithms)

This paper contains 20 sections, 35 theorems, 222 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. On the increasing degree property
  3. Critical pairs
  4. On the monomial staircase
  5. On the module of syzygies
  6. Proof of Theorem \ref{['thm_type1_good_pairs']}
  7. On the shape of the monomial staircase
  8. Gröbner bases of truncated polynomials
  9. The last dimension of the generic monomial staircase
  10. Algorithms and their properties
  11. Overall description of the F4 algorithm
  12. Descriptions of Algorithm \ref{['algo:F4B']} and Algorithm \ref{['algo:F4T']}
  13. Details on the reduction algorithm
  14. Structural properties
  15. Complexity analysis
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\bm{f} = (f_1, \ldots , f_n)$ be a sequence of nonzero polynomials in the ring $\mathcal{R}_{n}$, all of degree $\delta \geq 2$, with $n \geq 2$. Assume that $\delta$ is fixed. Let $\mathcal{T}$ be the Gröbner trace returned by Algorithm algo:F4B when executed with input $\bm{f}$ and the graded arithmetic operations in $\mathbb{K}$, for all $\varepsilon > 0$.

Figures (7)

  • Figure 1: exponential gain in the asymptotic
  • Figure 2: $\omega = 3$
  • Figure 3: $\delta = 10$
  • Figure 4: $\omega = 3$
  • Figure 5: $\delta = 10$
  • ...and 2 more figures

Theorems & Definitions (73)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of \ref{['prop_no_degree_fall']}
  • Definition 3.1
  • Theorem 3.2
  • ...and 63 more